Electronic Journal of Differential Equations, Vol. 2004(2004), No. 09, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)
COMMENTARY ON LOCAL AND BOUNDARY REGULARITY OF WEAK SOLUTIONS TO NAVIER-STOKES EQUATIONS ˇ SKALAK ´ ZDENEK
Abstract. We present results on local and boundary regularity for weak solutions to the Navier-Stokes equations. Beginning with the regularity criterion proved recently in [14] for the Cauchy problem, we show that this criterion holds also locally. This is also the case for some other results: We present two examples concerning the regularity of weak solutions stemming from the regularity of two components of the vorticity ([2]) or from the regularity of the pressure ([3]). We conclude by presenting regularity criteria near the boundary based on the results in [10] and [16].
1. Introduction Let Ω = R3 or Ω be a bounded domain in R3 with smooth boundary ∂Ω, let T > 0 and QT = Ω × (0, T ). Consider the Navier-Stokes initial-boundary value problem describing the evolution of the velocity u(x, t) and the pressure p(x, t) in QT : ∂u − ν∆u + u · ∇u + ∇p = 0 in QT , ∂t div u = 0 in QT , u = 0 on ∂Ω × (0, T ) if Ω is bounded, u = u0 , t=0
(1.1) (1.2) (1.3) (1.4)
where ν > 0 is the viscosity coefficient. The initial data u0 satisfy the compatibility conditions u0 |∂Ω = 0 and div u0 = 0. Let us stress that the condition (1.3) does not apply if Ω = R3 . As is usual in the standard theory of the Navier-Stokes equations, define D(Ω) = {ψ ∈ C0∞ (Ω)3 ; ∇ · ψ = 0 in Ω} and let L2σ (Ω) be the completion of D(Ω) in L2 (Ω)3 . Define also DT = {ϕ ∈ C0∞ (Ω × [0, T ))3 ; ∇ · ϕ = 0 in Ω × [0, T )}. Definition 1.1. Let u0 ∈ L2σ (Ω). A measurable function u : QT → R3 is called a weak solution of the problem (1.1)–(1.4) if u ∈ L2 (0, T, W 1,2 (Ω)) ∩ L∞ (0, T, L2σ (Ω)) 2000 Mathematics Subject Classification. 35Q35, 35B65. Key words and phrases. Navier-Stokes equation, weak solution, partial regularity, energy inequality. c
2004 Texas State University - San Marcos. Submitted August 26, 2003. Published January 13, 2004. Supported by Grant A2060302 from the Agency of AS CR, and by Project 5476 from the Institute of Hydrodynamics AS. 1
ˇ SKALAK ´ ZDENEK
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and Z
T
Z h
0
Ω
i ∂ϕ u· − ν∇u · ∇ϕ − u · ∇u · ϕ dxdt = − ∂t
Z u0 · ϕ(·, 0) dx Ω
for all ϕ ∈ DT . The existence of weak solutions is generally known but their uniqueness and regularity remains an open problem (see for example [18]). Leray-Hopf weak solutions of (1.1)–(1.4) are those weak solutions from Definition 1.1 which satisfy the energy inequality Z t ku(t)k22 + 2 k∇u(s)k22 ds ≤ ku0 k22 0
for every t ∈ (0, T ]. A condition stronger than the energy inequality is the so called strong energy inequality. We say that a weak solution of (1.1)–(1.4) satisfies the strong energy inequality, if Z t2 ku(t2 )k22 + 2 k∇u(s)k22 ds ≤ ku(t1 )k22 (1.5) t1
for every t2 ∈ (0, T ] and almost every 0 < t1 ≤ t2 . The pair (u, p) is called a suitable weak solutions of (1.1)–(1.4) if u is a weak solution of (1.1)–(1.4) from Definition 1.1 and together with the pressure p satisfy the so called generalized energy inequality, that is Z TZ Z TZ h ∂φ i 2 + ν∆φ + (|u|2 + 2p)u · ∇φ (1.6) 2ν |∇u| φ ≤ |u|2 ∂t 0 Ω 0 Ω for every non-negative real-valued function φ ∈ C0∞ (QT ). There is also an equivalent form of (1.6): Z Z tZ Z tZ ∂φ Z tZ |u|2 φ + 2ν |∇u|2 φ ≤ |u|2 + ν∆φ + (|u|2 u + 2pu) · ∇φ ∂t Ω×{t} 0 Ω 0 Ω 0 Ω (1.7) which holds for every non-negative real-valued function φ ∈ C0∞ (QT ) and every t ∈ (0, T ). The suitable weak solutions were thoroughly studied in [1] and used in [9, 11, 13]. Their existence is known under the assumption of a sufficient regularity of the initial condition u0 (see [1]). We use the suitable weak solutions in this paper in Theorems 2.2, 2.3 and 2.6. The main reason for their use is that the local boundary integrals which appear in the proofs of these theorems are easily controllable due to the boundedness of the velocity u and its derivatives and sufficient regularity of the pressure p near the boundary. It was proved in [6] that if the initial condition u0 is sufficiently smooth then there exists a suitable weak solution of (1.1)–(1.4) which satisfies the generalized energy inequality for every smooth test function. More precisely, the inequality Z Z t2 Z 2 |u| φ + 2ν |∇u|2 φ Ω×{t2 }
Z ≤ Ω×{t1 }
|u|2 φ +
Z
t1 t2
t1
Ω
Z Ω
|u|2
∂φ ∂t
Z + ν∆φ +
t2
t1
Z Ω
(1.8) (|u|2 u + 2pu)∇φ
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holds for every φ ∈ C ∞ (QT ), φ ≥ 0 and every 0 ≤ t2 < T and almost every 0 ≤ t1 ≤ t2 . By the choice of φ ≡ 1 on QT we see that such solutions satisfy the strong energy inequality which does not seem to be known for the suitable weak solutions satisfying the ”ordinary” generalized energy inequality (1.7). We will use these solutions in Theorem 3.2, where the assumption of the strong energy inequality is needed. Recall that a point (x0 , t0 ) ∈ Ω × (0, T ) is called a regular point of u if u is essentially bounded in a space-time neighbourhood U of (x0 , t0 ), that is if u ∈ L∞ (U ). A point (x0 , t0 ) ∈ Ω × (0, T ) is called singular if it is not regular. In this paper we use the following regularity criterion proved in [9, Theorem 2.2]. Let us present here only a simplified version for f ≡ 0. Theorem 1.2. Let (u, p) be a suitable weak solution of (1.1)–(1.4). Then there exists a positive number ε∗ with the following property. Assume that for a point z0 = (x0 , t0 ) ∈ QT the inequality Z 1 |∇u|2 < ε∗ (1.9) lim sup r→0 r Q(z0 ,r) holds, where Q(z0 , r) = B(x0 , r) × (t0 − r2 , t0 ). Then z0 is a regular point of u. In fact, Theorem 2.2 in [9] is still stronger than Theorem 1.2. It even says that the velocity u is H¨ older continuous function in some space-time neighborhood of z0 . The famous criterion proved in [1, Proposition 2], is weaker then Theorem 1.2, since it uses Q∗ (z0 , r) = B(x0 , r) × (t0 − 78 r2 , t0 + 18 r2 ) instead of Q(z0 , r). Let us note that an analogous result to Theorem 1.2 was proved in [15] for the case of x0 ∈ ∂Ω and ∂Ω ∩ Br (x0 ) lying in a plane for some r > 0. In what follows, we use the standard notation for the Lebesgue and Sobolev spaces (Lp and W k,p , respectively) and for the corresponding norms (k·kRp and k·kk,p , R respectively). To simplify the notation we often write f instead of Ω f (x) dx or R R f (x) dx instead of f (x, t) dx. We do not distinguish between (Lp )m and Lp . The outer normal vector is denoted by n. In the paper c stands for a generic constant. 2. local regularity The main goal of this section is to show that some criterions on regularity of Leray-Hopf weak solutions of the Cauchy problem for the Navier-Stokes equations hold also locally. We will present three examples of such local regularity results. The results are formulated for the suitable weak solutions. The boundary integrals appearing during the computations are then easily controllable due to the boundedness of the velocity u and all its space derivatives near the boundary. We will suppose in this section that D ⊂ QT is an open set and (x0 , t0 ) ∈ D is an arbitrary point. If (u, p) is a suitable weak solution of (1.1)–(1.4) and if one wants to show that (x0 , t0 ) is a regular point of u then it is possible to suppose that the following conditions are fulfilled (for a detailed discussion see [11, 7, 8, 12, 13]). There exist positive numbers ε1 < ε2 and τ such that B2 × [t0 − τ, t0 + τ ] ⊂ D and (B2 \ B1 ) × [t0 − τ, t0 + τ ] ∩ S(u) = ∅, where Bi = Bεi (x0 ), i = 1, 2 and S(u) is a set of all singular points of u from QT . Further, there are no singular points of u in B2 × [t0 − τ, t0 ), u and all its space derivatives are continuous in (B2 \ B1 ) × [t0 − τ, t0 + τ ] and ∂u ∂t and p and all their space derivatives are in
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Lβ ((B2 \ B1 ) × [t0 − τ, t0 + τ ]) for every β ∈ (1, 2). Moreover, if Ω = R3 then ∂u ∂t and p and all their space derivatives are in L∞ ((B2 \ B1 ) × [t0 − τ, t0 + τ ]). For further developments, we denote B3 = Bε3 (x0 ), where ε3 = (ε1 + ε2 )/2. As an inspiration for this section served the following regularity criterion proved recently by M. Pokorn´ y in [14]. Theorem 2.1. Let u0 ∈ W 1,2 (R3 ) with div u0 = 0, Ω = R3 and let u be a weak solution of the Navier-Stokes equations (1.1)–(1.4) satisfying the energy inequality. Assume moreover that ∇u3 ∈ Lα (0, T ; Lγ ) with α2 + γ3 ≤ 32 , 43 ≤ α ≤ ∞, 2 ≤ γ ≤ ∞. Then u and the corresponding pressure p is the smooth solution of the Navier-Stokes equations, i.e. u ∈ L∞ (0, T ; W 1,2 (R3 )) ∩ L2 (0, T ; W 2,2 (R3 )), ∇p ∈ L2 (0, T ; W 1,2 (R3 )). Moreover, u ∈ C ∞ ([δ, T ) × R3 ) and p ∈ C ∞ ([δ, T ) × R3 ) for any small positive δ. The following result is a local version of Theorem 2.1. Its proof uses the same ideas as the proof of Theorem 2.3 below. Theorem 2.2. Let (u, p) be a suitable weak solution of (1.1)–(1.4)) and D ⊂ QT be an open set. Assume moreover that ∇u3 ∈ Lα,γ (D) with α2 + γ3 ≤ 32 , 43 ≤ α ≤ ∞, 2 ≤ γ ≤ ∞. Then u has no singular points in D. ∂u3 ∂u3 ∂u1 ∂u1 ∂u2 2 Let ω = curl u = (ω1 , ω2 , ω3 ) = ( ∂u ∂x3 − ∂x2 , ∂x1 − ∂x3 , ∂x2 − ∂x1 ) denote the vorticity field. The two-component vorticity field is denoted ω ˜ = ω1 e1 +ω2 e2 , where e1 = (1, 0, 0), e2 = (0, 1, 0). The following theorem is a local version of Theorem 1 proved in [2].
Theorem 2.3. Let (u, p) be a suitable weak solution of (1.1))–(1.4). Let D ⊂ QT be an open set and ω ˜ ∈ Lα,γ (D) with α2 + γ3 ≤ 2, 1 < α < ∞, 32 < γ < ∞ or 3
the norm of ω ˜ in the space L∞, 2 (D) is sufficiently small. Then u has no singular points in D. Proof. We follow the lines of the proof in [2]. The boundary integrals can be handled because the suitable weak solution is considered. We begin with the equation ∂ω − ν∆ω + u · ∇ω − ω · ∇u = 0. (2.1) ∂t Multiplying (2.1) by ω and integrating it over B3 we get that Z Z Z 1 d ∂ω 1 2 2 kω(t)k2 + νk∇ω(t)k2 = ω · ∇u · ω + ·ω− u · n|ω|2 (2.2) 2 dt 2 ∂B3 B3 ∂B3 ∂n holds for almost every t ∈ (t0 − τ, t0 ). The last two boundary integral can be estimated by a constant c independent of time. R We estimate the integral B3 ω · ∇u · ω. We can express u by means of ω: Z Z 1 rot ω(ξ) 1 ∂u 1 u(x) = dξ + (ξ) dξ S 4π B2 |x − ξ| 4π ∂B2 ∂nξ |x − ξ| Z (2.3) 1 ∂ 1 − u(ξ) dξ. 4π ∂B2 ∂ξ |x − ξ| Equation (2.3) holds for every t ∈ (t0 −τ, t0 ) and every x ∈ B2 . Therefore, u = u ¯+u ¯, ¯ is defined where u ¯ is defined by the first integral on the right hand side of (2.3) and u by sum of the second and the third integral on the right hand side of (2.3). Since ¯ ∈ L∞ (B3 ×(t0 −τ, t0 )) for every multiindex γ = (γ1 , γ2 , γ3 ), γi ≥ 0, i = 1, 2, 3, Dxγ u
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¯ in the following procedures. it is possible to notice that there are no problems with u Therefore, we work only with u ¯ and for the sake of simplicity denote it as u. From the definition of u we have for every x ∈ B3 that Z ∂ui ∂ 1 (x) = lim ilk nk (ξ)ωl (ξ)dξ S ε→0+ ∂B (x) ∂ξj |x − ξ| ∂xj ε Z ∂ 1 − ilk nk (ξ)ωl (ξ)dξ S ∂ξ |x − ξ| j ∂B2 (2.4) Z 1 ∂2 ilk ωl (ξ)dξ + lim ε→0+ B (x) ∂ξj ∂ξk |x − ξ| 2,ε = I1 (x) + I2 (x) + I3 (x), where B2,ε (x) = B2 \ Bε (x) and ijk is the Levi-Civita’s tensor. After short computation one can get 4π I1 (x) = − ijl ωl (x) 3 R and if we put I1 (x) into the integral B3 ω · ∇u · ω instead of ∇u we get Z Z 4π ω · ∇u · ω = − ijl ωi ωj ωl = 0. (2.5) 3 B3 B3 Since |x − ξ| > (ε2 − ε1 )/2 for every x ∈ B3 and every ξ ∈ ∂B2 , I2 is bounded in B3 × (t0 − τ, t0 ) and for the sake of simplicity we do not consider this term any further. R If we put I3 (x) into the integral B3 ω · ∇u · ω instead of ∇u and decompose ˜ ˜ ω=ω ˜ +ω ˜, ω ˜ = (0, 0, ω3 ), we get Z Z ∂2 1 ωj (x) · lim ilk ωl (ξ)dξ · ωi (x) dx ε→0+ B (x) ∂ξj ∂ξk |x − ξ| B3 2,ε Z (2.6) = ωj Pij (ω)ωi B Z 3 Z Z Z ˜˜ i + ˜˜ )˜ ˜˜ )ω ˜˜ i . = ωj Pij (˜ ω )˜ ωi + ωj Pij (˜ ω )ω ωj Pij (ω ωi + ωj Pij (ω B3
B3
B3
B3
(Pij (·))3i,j=1
where P (·) = denotes the singular integral operator defined by the third integral in (2.4). The last integral is equal to zero. The remaining three integrals on the right hand side of (2.6) can be estimated by Z Z Z ˜ ˜˜ )||˜ |ω||P (˜ ω )||ω ˜| + |ω||P (˜ ω )||˜ ω| + |ω||P (ω ω| B3 B3 B3 Z Z (2.7) 2 ˜ ≤c |ω| |P (˜ ω )| + c |ω||P (ω ˜ )||˜ ω | = J1 + J2 . B3
B3
We have by the H¨ older inequality, the Calderon-Zygmund inequality, the interpolation inequality, the Sobolev inequality and the Young inequality that 2γ−3 γ
J1 ≤ kP (˜ ω )kγ kωk22γ ≤ ck˜ ω kγ kωk2 γ−1
3
k∇ωk2γ ≤
1 2 νk∇ωk22 + Ck˜ ω kα γ kωk2 . (2.8) 4
The same estimate can be obtained for J2 : 1 2 J2 ≤ νk∇ωk22 + Ck˜ ω kα γ kωk2 . 4
(2.9)
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If α = ∞ and γ = 3/2 then both J1 and J2 can be estimated by Ck˜ ω k3/2 k∇ωk22 .
(2.10)
We get from (2.2)–(2.10) that d 2 kω(t)k22 + νk∇ω(t)k22 ≤ Ck˜ ω kα γ kωk2 + C dt and by the Gronwall lemma we have ω ∈ L∞ (t0 − τ, t0 ; L2 (B3 )) ∩ L2 (t0 − τ, t0 ; W 1,2 (B3 )).
(2.11)
(2.12)
We can write for every x ∈ B3 that Z Z 1 1 1 1 ∂u u(x) = ∇ξ × ω(ξ) dξ + ω(ξ) × n(ξ) + (ξ) dξ S 4π B2 |x − ξ| 4π ∂B2 |x − ξ| ∂n Z ∂ 1 1 − u(ξ) dξ S. 4π ∂B2 ∂nξ |x − ξ| (2.13) The boundary integrals on the right hand side of (2.13) cause no problems because they are from L∞ (t0 − τ, t0 ; L∞ (B3 )). Applying now the famous results (see e.g. [5]) on the first integral in (2.13) we get that u ∈ L∞ (t0 − τ, t0 ; L6 (B3 )) and from this we are going to derive that u ∈ L∞ (t0 − τ, t0 ; W 1,2 (B3 )).
(2.14)
In fact we will show a stronger result which we will need further in the proof of Theorem 2.4: if u ∈ L∞ (t0 −τ, t0 ; Ls (B3 )), s ∈ (3, 6], then u ∈ L∞ (t0 −τ, t0 ; W 1,2 (B3 )). Let us multiply the equation (1.1) by −∆u, integrate it over B3 and use the integration by parts. We get Z Z Z ∂u 1 d k∇uk22 + νk∆uk22 = u · ∇u · ∆u + p n · ∆u + n · ∇u · . (2.15) 2 dt ∂t B3 ∂B3 ∂B3 β The last two boundary R integrals are from L (t0 − τ, t0 ) for any β ∈ (1, 2). Let us estimate the integral B3 u · ∇u · ∆u. Z 2s k∆uk2 u · ∇u · ∆u ≤ kuks k∇uk s−2 B3
s−3
3
≤ kuks k∇uk2 s k∇uk6s k∆uk2 s−3 s
2
s+3 s
(2.16)
≤ ckuks k∇uk2 k∇ uk2 2s ν ≤ k∇2 uk22 + ckukss−3 k∇uk22 , 2 where we used the H¨ older inequality, the interpolation inequality, the Sobolev inequality and the Young inequality. (2.14) now follows from (2.15) and (2.16). It means that the assumption (1.9) is fulfilled and Theorem 1.2 gives that (x0 , t0 ) is a regular point of u. Since (x0 , t0 ) was an arbitrary point in D, Theorem 2.3 is proved. We will now turn our attention to the paper [3]. The main result of this paper is the following theorem:
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Theorem 2.4. Let u0 ∈ L2 (R3 ) ∩ Lq (R3 ) for some q > 3 with div u0 = 0 in the sense of distributions and Ω = R3 . Suppose that u is a Leray-Hopf weak solution of (1.1)–(1.4) in [0, T ). If p ∈ Lα,γ (R3 ) with α2 + γ3 < 2 and 1 < α ≤ ∞, 32 < γ < ∞, or p ∈ L1,∞ , or else kpkL∞,3/2 is sufficiently small, then u is a regular solution in [0, T ). For some α, γ it is possible to prove a local version of Theorem 2.4 - see Theorem 2.6. In the proof of Theorem 2.6 we will use the following Gronwall lemma (see [17, p. 88]). Lemma 2.5. Let g, h and y be locally integrable nonnegative functions on [0, ∞) that satisfy the differential inequality y 0 (t) ≤ g(t)y(t) + h(t)
on [0, ∞), y(0) = y0 .
Let the function y 0 (t) be also locally integrable. Then Z t Z t Z y(t) ≤ y(0) exp g(τ ) dτ + h(s) exp − 0
0
s
g(τ ) dτ ds, t ≥ 0.
t
Theorem 2.6. Let (u, p) be a suitable weak solution of (1.1))–1.4. Let D ⊂ QT is an open set and p ∈ Lα,γ (D) with 1 ≤ α ≤ ∞, 23 ≤ γ ≤ ∞. Then u has no singular points in D if one of the following conditions is fulfilled: (i) α2 + γ3 < 2, γ ∈ (3, ∞) and α ≥ 2 (ii) α2 + γ3 < 2, γ ∈ (3, ∞) and Ω = R3 (iii) α2 + γ3 < 2 and γ ∈ ( 32 , 3] (iv) γ = ∞, α = 1 and Ω = R3 , (v) γ = 32 , α = ∞ and the norm kpk∞, 32 is sufficiently small. Proof. Let s > 3. The proof is based on the inequality Z Z Z s−2 d |u|s + 2 |∇|u|s/2 |2 ≤ 2(s − 2) |p||u| 2 |∇|u|s/2 | dt B3 B3 B3 Z + spu · n|u|s−2 + boundary integrals, ∂B3
(2.17) which can be obtained multiplying the equation (1.1) by su|u|s−2 and using the integration by parts. The boundary integrals and the boundary integral with p on the right hand side of (2.17) are from the space L1 (t0 − τ, t0 ). Using the Gronwall lemma they play the role of the function h. We can write Z 1 ∂2 1 1 p(x) = lim ui (y)uj (y) dy − |u(x)|2 ε→0+ 4π B (x) ∂y ∂y |x − y| 3 i j 2,ε Z n 1 ∂ 1 ∂ 1 + (ui (y)uj (y))ni (y) − ui (y)uj (y)nj (y) 4π ∂B2 ∂yj |x − y| ∂yi |x − y| ∂p 1 ∂ 1 o + (y) − p(y) dy S ∂n |x − y| ∂ny |x − y| = p1 (x) + p2 (x), (2.18) for every x ∈ B3 , where p2 is defined by the boundary integral in (2.18). Let us note that in the following estimates p1 will be handled using the Calderon-Zygmund
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inequality and p2 ∈ Lβ (t0 − τ, t0 ; L∞ (B3 )) for every β ∈ (1, 2) if Ω is a bounded domain and p2 ∈ L∞ (t0 − τ, t0 ; L∞ (B3 )) if Ω = R3 . R s−2 Let us estimate the integral I = B3 |p||u| 2 |∇|u|s/2 | on the right hand side of (2.17). We will discus the assumptions (i)–(v). (i) α2 + γ3 < 2, γ ∈ (3, ∞) and α ≥ 2. Then there exists s ∈ (3, γ), such that 2 3 s−3 − ≥ . α γ γ
2−
(2.19)
By the H¨ older inequality and the Young inequality we have s−2
s
I ≤ kpks kuks 2 k∇|u| 2 k2 s
s(1− 2s )
s
s(1− 2s )
≤ k∇|u| 2 k22 + ckpk2s kuks
≤ k∇|u| 2 k22 + ckpkα γ kuks
(2.20) .
Without loss of generality we can suppose that kuks > 1 for almost every t ∈ s (t0 − τ, t0 ). The inequality (2.20) then gives that I ≤ k∇|u|s/2 k22 + ckpkα γ kuks and by the use of (2.17) and Lemma 2.5 we get that u ∈ L∞ (t0 − τ, t0 ; Ls (B3 )).
(2.21)
We will show that (2.21) holds also in the case of conditions (ii)–(v). (ii) α2 + γ3 < 2, γ ∈ (3, ∞) and Ω = R3 . Again, as in (i), there exists s ∈ (3, γ) such that (2.19) is satisfied. Then s−2
γ−s
s
γ
s−2
s
I ≤ kpks kuks 2 k∇|u| 2 k2 ≤ kpk 2γ−s kpkγ2γ−s kuks 2 k∇|u| 2 k2 ≤ c(I1 + I2 ), (2.22) s 2
where we used the interpolation inequality and γ−s
γ
s−2
s
Ik = kpk k 2γ−s kpkγ2γ−s kuks 2 k∇|u| 2 k2 , s 2
k = 1, 2.
Due to the definition of p1 in (2.18) and the Calderon-Zygmund inequality we have 2γ−2s
I1 ≤ kuks2γ−s
+ s−2 2
γ
2 s(1− 2γ−s )
s
≤ k∇|u| 2 k22 + ckuks since
2γ 2γ−s
s
2 s(1− 2γ−s )
s
kpkγ2γ−s k∇|u| 2 k2 ≤ k∇|u| 2 k22 + ckuks
2γ
kpkγ2γ−s
(2.23)
kpkα γ,
≤ α as a consequence of the inequality (2.19). Furthermore, 2γ
s
2γ−2s
I2 ≤ k∇|u| 2 k22 + ckuks−2 kpkγ2γ−s kp2 k s2γ−s . s 2
(2.24)
Since Ω = R3 , we know that p2 ∈ L∞ (t0 − τ, t0 ; L∞ (B3 )) and s
2γ
s
s(1− 2s )
kpkγ2γ−s ≤ k∇|u| 2 k22 + ckuks I2 ≤ k∇|u| 2 k22 + ckuks−2 s
kpkα γ.
(2.25)
In the same way as in (i) we can conclude from (2.17), (2.22), (2.23), (2.25) and Lemma 2.5 that (2.21) is satisfied. (iii) α2 + γ3 < 2 and γ ∈ ( 32 , 3]. Let us suppose firstly that γ ∈ (9/4, 3]. Then there exists s > 3 such that 3s 2 3 3 γ> and 2 − − ≥ 1 − . s+1 α γ s
(2.26)
(2.27)
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The H¨ older inequality, the interpolation inequality, the Sobolev inequality and the Young inequality give s−2
γs−3s+γ 2γ
s
2 I ≤ kpkγ kuk γ(s−2) k∇|u| 2 k2 ≤ kpkγ kuks
3s−3γ
s
kuk3s2γ k∇|u| 2 k2
γ−2 γs−3s+γ 2γ
≤ kpkγ kuks
3s−3γ γs
s
k∇|u| 2 k2
+1
2γ ) s(1− γs−3s+3γ
s
2γ s(1− γs−3s+3γ )
2γs
s
≤ k∇|u| 2 k22 + ckpkγγs−3s+3γ kuks
≤ k∇|u| 2 k22 + ckpkα γ kuks
.
(2.28) 2γs The last inequality follows from γs−3s+3γ ≤ α (which is a consequence of the second inequality in (2.27)) and (2.21) is satisfied. Secondly, let γ ∈ (3/2, 9/4]. (2.29) Then there exists s > 3 such that 2 3 s−3 . (2.30) 2− − > α γ α The H¨ older inequality and the interpolation inequality give s−2
γ(s−1)
s
3s−γ−γs
s−2
s
3s−2γ 2 2 3s kuk I ≤ kpk s+1 kpk 3s3s−2γ kuk3s2 k∇|u| 2 k2 ≤ c(I1 + I2 ), 3s k∇|u| k2 ≤ kpkγ 2 (2.31) where γ(s−1)
3s−γ−γs
s−2
s
Ik = kpkγ3s−2γ kpk k 3s3s−2γ kuk3s2 k∇|u| 2 k2 ,
k = 1, 2.
2
By the H¨ older inequality, the Sobolev inequality, the Young inequality and the fact that γs−γ 2γ−3 ≤ α (which follows from the inequality (2.30)) we have γ(s−1)
6s−2γ−2γs
I1 ≤ kpkγ3s−2γ kuk3s 3s−2γ γ(s−1)
+ s−2 2
γ(s−1)
s
s
6s−8γ+6
k∇|u| 2 k2 ≤ kpkγ3s−2γ k∇|u| 2 k2 3s−2γ
s
(2.32)
s
2 2 ≤ ckpkγ2γ−3 + k∇|u| 2 k22 ≤ ckpkα γ + k∇|u| k2 .
Furthermore, by the Sobolev inequality and the Young inequality γ(s−1)
3s−γ−γs
s
s−2
+1
γs(s−1)
s
s(3s−γ−γs)
≤ k∇|u| 2 k22 + ckpkγ3s−2γ kp2 k 3s 3s−2γ . 2 2 (2.33) Now we show the integrability in time of the second part of the right hand side of the last inequality. The H¨ older inequality gives for some β ∈ (1, 2) sufficiently close to 2 that Z t0 s(3s−γ−γs) γs(s−1) s(3s−γ−γs) γs(s−1) kp2 kβ, 3s3s−2γ . (2.34) kpkγ3s−2γ kp2 k 3s 3s−2γ dt ≤ kpk 3s−2γβsγ(s−1) I2 ≤ kpkγ3s−2γ kp2 k 3s3s−2γ k∇|u| 2 k2 s
t0 −τ
2
β(3s−2γ)−s(3s−γ−sγ)
,γ
2
s(3s−γ−γs) 3s−2γ β, 3s 2
Since p2 ∈ Lβ (t0 − τ, t0 ; L∞ (B3 )), we have kp2 k < ∞. To show the boundedness of the integral on the left hand side of the inequality (2.34), it is now sufficient to realize that βsγ(s − 1) ≤ α. (2.35) β(3s − 2γ) − s(3s − γ − sγ) We know that
2γ < α. 2γ − 3
(2.36)
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If we denote gγ (β, s) =
βsγ(s − 1) , β(3s − 2γ) − s(3s − γ − sγ)
(2.37)
then for every γ ∈ ( 23 , 94 ], gγ is a continuous function in β, s defined in an open neighbourhood of the point (2, 3). It can be easily verified that 3 9i 2γ gγ (2, 3) ≤ for every γ ∈ , . (2.38) 2γ − 3 2 4 For β sufficiently close to 2 and s sufficiently close to 3 the inequality (2.35) now follows from (2.36), (2.37), (2.38) and the continuity of gγ at the point (2, 3). Therefore, the value of the integral on the left hand side of the inequality (2.34) is less than ∞ and (2.21) now follows from (2.17), (2.31), (2.32), (2.33) and Lemma 2.5. (iv) γ = ∞, α = 1 and Ω = R3 . Then by the H¨older inequality 1
s−2
1
s
2 I ≤ kpk∞ kpk 2s kuks 2 k∇|u| 2 k2 ≤ c(I1 + I2 ),
(2.39)
2
where 1
s−2
1
s
2 Ik = kpk∞ kpk k 2s kuks 2 k∇|u| 2 k2 ,
k = 1, 2.
2
In a standard way we can estimate 1
s
s
s
2 I1 ≤ kpk∞ kuks2 k∇|u| 2 k2 ≤ ckpk∞ kukss + k∇|u| 2 k22 ,
s(1− 2s )
s
I2 ≤ k∇|u| 2 k22 + ckpk∞ kp2 k 2s kuks
.
(2.40) (2.41)
∞
∞
The term kpk∞ kp2 ks/2 above is integrable in time since p2 ∈ L (t0 −τ, t0 ; L (B3 )) and (2.21) now follows from (2.39), (2.40), (2.41), (2.17) and Lemma 2.5. (v) γ = 32 , α = ∞ and the norm kpk∞, 32 is sufficiently small. Then for s > 3 s−2
1
s
1
s−2
s
2 2 2 2 2 2 3s kuk I ≤ kpk s+1 3s k∇|u| k2 ≤ kpk 3 kpk 3s kuk3s k∇|u| k2 ≤ c(I1 + I2 ), 2
(2.42)
2
where 1
s−2
1
s
Ik = kpk 23 kpk k 23s kuk3s2 k∇|u| 2 k2 , 2
k = 1, 2.
2
The Calderon-Zygmund inequality gives 1
s
1
s
s
2 I1 ≤ kpk 23 kuk3s k∇|u| 2 k2 ≤ kpk 23 k∇|u| 2 k22 2
(2.43)
2
and we see that to finish the proof in this case the sufficient smallness of the norm kpk∞, 32 is necessary. Furthermore, 1
1
s−2
s
1
1
s
2(s−1) s
I2 = kpk 23 kp2 k 23s kuk3s2 k∇|u| 2 k2 ≤ kpk 23 kp2 k 23s k∇|u| 2 k2 2
≤ k∇|u| s/2
2
s 2
k22
2
s 2 3 2
s 2 3s 2
2
(2.44)
+ ckpk kp2 k .
s
The term kpk 3 kp2 k 23s is clearly integrable in time if s ∈ (3, 4). Let us notice 2 2 that for the estimate of I2 we did not need the assumption on the smallness of the norm kpk∞, 32 . Again, (2.21) now follows from (2.17), (2.42), (2.43), (2.44) and Lemma 2.5. Thus, (2.21) holds for every condition (i) - (v) for some s > 3. As was shown in the proof of Theorem 2.3, we then have u ∈ L∞ (t0 − τ, t0 ; W 1,2 (B3 )) and the proof of Theorem 2.6 can be concluded by the use of Theorem 1.2.
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LOCAL AND BOUNDARY REGULARITY
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3. Boundary regularity In this section Ω is a bounded domain in R3 with smooth boundary ∂Ω. For r > 0 we denote Ur = Ur (∂Ω) = {x ∈ Ω; dist(x, ∂Ω) < r}. In [10] J. Neustupa proved the following result. Theorem 3.1. Let u be a weak solution of (1.1)–(1.4) that satisfies the strong energy inequality. Let 0 ≤ t1 < t2 ≤ T and one of the two following conditions be fulfilled: ∗ (i) u ∈ Lp (t1 , t2 ; Lq (Ur )3 ) for some r > 0, p2 + q3∗ ≤ 1, p ∈ [2, ∞], q ∗ ∈ (3, ∞], (ii) u ∈ L∞ (t1 , t2 ; L3 (Ur )3 ) and kukL∞ (t1 ,t2 ;L3 (Ur )3 ) is sufficiently small. Let ζ > 0 be such that t1 + ζ < t2 − ζ. Then u ∈ L∞ (t1 + ζ, t2 − ζ; W 2+δ,2 (Uρ )3 ) ∞ δ,2 and both ∂u (Uρ )3 ) for each δ ∈ [0, 12 ) and ∂t and ∇p belong to L (t1 + ζ, t2 − ζ; W ρ ∈ (0, r). It follows from Theorem 3.1 that u ∈ L∞ (Uρ × (t1 + ζ, t2 − ζ)) and there are no singular points near or on the boundary ∂Ω. We show that the same result as in Theorem 3.1 can be proved if the conditions (i), (ii) are replaced by conditions on the vorticity field ω. To this purpose we are going to work with the suitable weak solutions which satisfy the generalized energy inequality for every smooth test function and were discussed in Introduction. We will prove the following theorem. Theorem 3.2. Let (u, p) be a suitable weak solution of (1.1)–(1.4) that satisfies the generalized energy inequality (1.8) for every φ ∈ C ∞ (QT ), φ ≥ 0 and every 0 < t1 ≤ t2 < T . Let ω ∈ Lp (t1 , t2 ; Lq (UR )) (3.1) 3 2 for some R > 0, p + q ≤ 2, p ∈ [2, ∞], q ∈ [ 32 , 3]. If p = ∞ and q = 32 we still 3
suppose that the norm of ω in L∞ (t1 , t2 ; L 2 (UR )) is sufficiently small. Then either the condition (i) or the condition (ii) from Theorem 3.1 is fulfilled for any 0 < r < R and thus all the conclusions of Theorem 3.1 hold. Especially, there exist no singular points of u in U r × (t1 − ζ, t2 + ζ) for any r ∈ (0, R) and ζ > 0. Proof. Let 0 < r < η < R. We start with the equality (2.13) with Uη instead of B2 which holds for every x ∈ Ur . Moreover, η can be chosen in such a way that there are no singular points of u on (∂Uη ∩ Ω) × (0, T ). It follows from the fact that u is a suitable weak solution - for a detailed discussion see [11], [12], [13]. The boundary of Uη consists of two parts, ∂Ω and ∂Uη ∩ Ω. Let us investigate firstly the boundary integrals in (2.13) over ∂Ω. The second integral is equal to zero due to the homogeneous boundary conditions (1.3). After short computation the first integral can be written as Z 1 ∇u(ξ) · n(ξ) dξ S. (3.2) |x − ξ| ∂Ω Let ξ ∈ ∂Ω is an arbitrary point. Then ∂uj ∂xi (ξ)nj (ξ)
∂uj ∂xi (ξ)nk (ξ)
is a tensor of the third order
and therefore is a vector. Let us choose a new coordinate system with the x1 axis in the direction of the outer normal vector to ∂Ω in the point ξ. Then ∂u 1 the vector ∂xji (ξ)nj (ξ) has the form ( ∂u ∂x1 , 0, 0) due to the homogeneous boundary
ˇ SKALAK ´ ZDENEK
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conditions (1.3) and the fact that in the new coordinate system n2 (ξ) = n3 (ξ) = 0. Using the continuity equation (1.2) and the homogeneous boundary conditions (1.3) ∂uj ∂u2 ∂u3 1 we obtain ∂u ∂x1 (ξ) = − ∂x2 (ξ) − ∂x3 (ξ) = 0. Thus, ∂xi (ξ)nj (ξ) is a zero vector in any coordinate system. We can conclude that the integral (3.2) is equal to zero and the equality (2.13) can be written as Z 1 1 u(x) = ∇ξ × ω(ξ) dξ 4π Uη |x − ξ| Z 1 1 ∂u + ω(ξ) × n(ξ) + (ξ) dξ S (3.3) 4π ∂Uη ∩Ω |x − ξ| ∂n Z 1 ∂ 1 − u(ξ) dξ S = u1 (x) + u2 (x). 4π ∂Uη ∩Ω ∂nξ |x − ξ| Since u ∈ C ∞ (Ω) for almost every t ∈ (0, T ) (see e.g. [4], Theoreme de Structure), the equality (3.3) holds for almost every t ∈ (t1 , t2 ). 1 1 ≤ η−r for every x ∈ Ur and every ξ ∈ ∂Uη ∩Ω and u is bounded Now, since |x−ξ| on (∂Uη ∩ Ω) × (t1 , t2 ), we have that u2 ∈ L∞ (Ur × (t1 , t2 )).
(3.4)
For the first integral in (3.3) we use the result from [5], Lemma 7.12, and get ∗ that u1 ∈ Lp (t1 , t2 ; Lq (Ur )), where 1q − q1∗ = 13 and p2 + q3∗ = 1. Therefore, if 3 ∗ 2 < q ≤ 3 and 3 < q ≤ ∞, then u1 and also u (see (3.4)) satisfy the condition (i) from Theorem 3.1. If q = 23 and q ∗ = 3, then u1 satisfies the condition (ii) from Theorem 3.1. The proof of Theorem 3.2 can now be concluded by the use of (3.4) and Theorem 2.3. In the final part of this paper, we discuss the paper [16], in which the following result on the boundary regularity of weak solutions was proved. Theorem 3.3. Let u be a weak solution of (1.1)–(1.4), x0 ∈ ∂Ω, 0 < t1 ≤ t2 < T and δ > 0. We denote Ω1 = Bδ (x0 ) ∩ Ω and suppose that ∂Ω1 ∩ ∂Ω is a part of a plane. Let one of the two following conditions be fulfilled: ∗ (i) u ∈ Lp (t1 , t2 ; Lq (Ω1 )), p2 + q3∗ ≤ 1, p ∈ [2, ∞], q ∗ ∈ (3, ∞] (ii) u ∈ L∞ (t1 , t2 ; L3 (Ω1 )) and kukL∞ (t1 ,t2 ;L3 (Ω1 )) is sufficiently small. Let ζ > 0 be such that t1 + ζ < t2 − ζ. Then u has no singular points in (Bδ0 (x0 ) ∩ Ω) × (t1 + ζ, t2 − ζ) for every δ 0 ∈ (0, δ). As for Theorem 3.2, we prove the following version of Theorem 3.3, involving the conditions on the vorticity ω. Theorem 3.4. Let u be a weak solution of (1.1)–(1.4) that satisfies the strong energy inequality (1.5), x0 ∈ ∂Ω, 0 < t1 ≤ t2 < T and δ > 0. We denote Ω1 = Bδ (x0 ) ∩ Ω and suppose that ∂Ω1 ∩ ∂Ω is a part of a plane. Let ω ∈ Lp (t1 , t2 ; Lq (Ω1 )) with
2 p
+
3 q
and
∇u ∈ Lp (t1 , t2 ; L1 (Ω1 )),
≤ 2, p ∈ [2, ∞], q ∈ [ 32 , 3]. If p = ∞ and q = ∞
3 2
3 2
(3.5)
we still suppose that the
norm of ω in L (t1 , t2 ; L (Ω1 )) is sufficiently small. Let ζ > 0 be such that t1 + ζ < t2 − ζ. Then u has no singular points in (Bδ0 (x0 ) ∩ Ω) × (t1 + ζ, t2 − ζ) for every δ 0 ∈ (0, δ).
EJDE-2004/09
LOCAL AND BOUNDARY REGULARITY
13
Proof. We proceed as in the proof of Theorem 3.2. Given 0 < δ 0 < δ there exist η1 , η = η(t) and δ 00 such that 0 < δ 0 < δ 00 < η1 < η < δ. We start with the equality (2.13) with B2 replaced by Bη (x0 ) ∩ Ω which holds for every x ∈ Bδ00 (x0 ) ∩ Ω. In the same way as in the proof of Theorem 3.2 we get that the boundary integrals over ∂Ω are equal to zero and we have Z 1 1 u(x) = ∇ξ × ω(ξ) dξ 4π Bη (x0 )∩Ω |x − ξ| Z 1 ∂u 1 + ω(ξ) × n(ξ) + (ξ) dξ S (3.6) 4π ∂Bη (x0 )∩Ω |x − ξ| ∂n Z 1 ∂ 1 − u(ξ) dξ S = u1 (x) + u2 (x), 4π ∂Bη (x0 )∩Ω ∂nξ |x − ξ| which holds for every x ∈ Bδ00 (x0 ) ∩ Ω and almost every t ∈ (t1 , t2 ). Now, u1 can be treated in the same way as in Theorem 3.2 and we get that u1 fulfills the conditions (i) (if q > 32 ) or (ii) (if p = ∞ and q = 32 ) from Theorem 3.3, where 1q − q1∗ = 13 . Let us discuss u2 . Let t ∈ (t1 , t2 ). There exists η = η(t) ∈ (η1 , δ) such that Z Z 1 |∇u| dx S ≤ |∇u| dx. (3.7) δ − η1 (Bδ (x0 )\Bη1 (x0 ))∩Ω ∂Bη (x0 )∩Ω Thus, for every x ∈ Bδ00 (x0 ) ∩ Ω we have due to the definition u2 and (3.7) that Z Z |u2 (x)| ≤ c |∇u| dx S ≤ c |∇u| dx (3.8) ∂Bη (x0 )∩Ω
Ω1
and ku2 kL∞ (Bδ00 (x0 )∩Ω) ≤ ck∇ukL1 (Ω1 ) . It follows from (3.5) and (3.9) that u2 ∈ Lp (t1 , t2 ; L∞ (Bδ00 (x0 ) ∩ Ω))
(3.9) (3.10)
and the conclusion of Theorem 3.4 follows from (3.10), the facts proved above on u1 and Theorems 3.3, and 2.3. References [1] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. on Pure and Appl. Math. 35 (1982), 771–831. [2] D. Chae and H.J. Choe, Regularity of solutions to the Navier-Stokes equations, Electronic J. of Diff. Equations 5 (1999), 1–7. [3] D. Chae and J. Lee, Regularity criterion in terms of pressure for the Navier-Stokes equations, University of Minnesota Preprint Series 1614 (1999), 1–9. [4] G.P. Galdi, An Introduction to the Navier-Stokes initial-boundary value problem, in Fundamental Directions in Mathematical Fluid Mechanics, editors G.P. Galdi, J. Heywood and R. Rannacher, series ”Advances in Mathematical Fluid Mechanics”, Birkhauser-Verlag, Basel (2000), 1–98. [5] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin Heidelberg (2001). [6] P. Kuˇ cera, Z. Skal´ ak, Generalized energy inequality for suitable weak solutions of the NavierStokes equations, Proceedings of Seminar ”Topical Problems of Fluid Mechanics 2003”, Institute of Thermomechanics AS CR, Editors: J.Pˇrihoda, K.Kozel, Prague, (2003), 61-66. [7] P. Kuˇ cera, Z. Skal´ ak, Regularity of suitable weak solutions of the Navier-Stokes equations as consequence of regularity of two velocity components, Proceedings of Seminar ”Topical Problems of Fluid Mechanics 2002”, Institute of Thermomechanics AS CR, Editors: K.Kozel, J.Pˇrihoda, Prague, (2002), 53-56.
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[8] P. Kuˇ cera, Z. Skal´ ak, Smoothness of the velocity time derivative in the vicinity of regular points of the Navier-Stokes equations, Proceedings of the 4th Seminar ”Euler and NavierStokes equations (Theory, Numerical Solution, Applications)”, Institute of Thermomechanics AS CR, Editors: K.Kozel, J.Pˇrihoda, M.Feistauer, Prague, (2001), 83-86. [9] O.A. Ladyzhenskaya, G.A. Seregin, On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech 1 (1999), 356–387. [10] J. Neustupa, The boundary regularity of a weak solution of the Navier-Stokes equation and connection with the interior regularity of pressure, it will be published in Applications of Mathematics, 1–10. [11] J. Neustupa, A. Novotn´ y and P. Penel, A interior regularity of a weak solution to the NavierStokes equations in dependence on one component of velocity, to apper in Topics in Mathematical Fluid Mechanics, a special issue of Quaderni di Matematica. [12] J. Neustupa, P. Penel, Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier-Stokes equations, in Mathematical Fluid Mechanics, Recent Results and Open Problems, editors J. Neustupa and P. Penel, series ”Advances in Mathematical Fluid Mechanics”, Birkhauser-Verlag, Basel, (2001), 237–268. [13] J. Neustupa, P. Penel, Regularity of a suitable weak solution to the Navier-Stokes equations as a consequence of regularity of one velocity component, Nonlinear Applied Analysis, editor A. Sequeira, Plenum Press, New-York, (1999), 391–402. [14] M. Pokorn´ y, On the result of He concerning the smoothness of solutions to the Navier-Stokes equations, Electronic J. of Diff. Equations 2003 (2003), No. 11, 1–8. [15] G.A. Seregin, Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary, J. Math. Fluid Mech. 4 (2002), 1–29. [16] S. Takahashi, On a regularity criterion up to the boundary for weak solutions of the NavierStokes equations, Comm. on Partial Differential Equations 17 (1992), 261–283. [17] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, SpringerVerlag New York Inc., (1988). [18] R. Temam, Navier-Stokes equations, theory and numerical analysis, North@-Holland Publishing Company, Amsterodam, New York, Oxford, (1979). ˇankou 30/5, 166 12 Institute of Hydrodynamics, Czech Academy of Sciences, Pod Pat Prague 6, Czech Republic E-mail address:
[email protected],
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